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<b>let </b><font color="Maroon">S</font> be  non <a href="pencil_1.html#V3">void</a> <a href="pencil_1.html#V6">identifying_close_blocks</a>  <a href="pre_topc.html#L1">TopStruct</a> ;<br/>

<b>assume </b><a NAME="E1:3"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">S</font> is <a href="pencil_1.html#V10">strongly_connected</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>;<br/><b>consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:3_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</font> as   <a href="pencil_1.html#NM1">Block</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">X1</font> = <font color="Maroon">X</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/><a NAME="E5:3_1"/>
( <font color="Maroon">X1</font> is <a href="pencil_1.html#V1">closed_under_lines</a> &amp; <font color="Maroon">X1</font> is <a href="pencil_1.html#V2">strong</a> )
 <b>by </b><i><a class="ref" href="pencil_1.html#T20">PENCIL_1:20</a>, <a class="ref" href="pencil_1.html#T21">PENCIL_1:21</a></i>;<br/><b>then consider </b><font color="Maroon">f</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E7:3_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> &amp; ( for <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>  st <font color="Olive">W</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> holds <br/>( <font color="Olive">W</font> is <a href="pencil_1.html#V1">closed_under_lines</a> &amp; <font color="Olive">W</font> is <a href="pencil_1.html#V2">strong</a> ) ) &amp; ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> holds <br/>2 <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> ) )
 <b>by </b><i>, <a class="ref" href="pencil_1.html#D11">PENCIL_1:def 11</a></i>;<br/><a NAME="E8:3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E9:3_1"/><b>then </b><i><font color="Green">E47</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/><a NAME="E10:3_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>then reconsider </b><font color="Maroon">W</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> ;<br/><a NAME="E12:3_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">W</font> is <a href="pencil_1.html#V1">closed_under_lines</a> &amp; <font color="Maroon">W</font> is <a href="pencil_1.html#V2">strong</a> &amp; ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> holds <br/>2 <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> ) )
 <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">l</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i>, <a class="ref" href="finseq_3.html#T28">FINSEQ_3:28</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:3_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> or  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_1"><b>suppose </b></a><a NAME="E1:3_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:3_1_1_1"/>
 ex <font color="Olive">l</font> being  <a href="pencil_1.html#NM1">Block</a> of <font color="Maroon">S</font> st <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">l</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_2"><b>suppose </b></a><a NAME="E1:3_1_1_2"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><div class="add"><a NAME="E2:3_1_1_2"/><b>then </b><i><font color="Green">E51</font></i>: 
1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/><a NAME="E3:3_1_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:3_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E5:3_1_1_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:3_1_1_2"/><b>then </b>
2 <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:3_1_1_2"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="pencil_1.html#T3">PENCIL_1:3</a></i>;<br/><a NAME="E9:3_1_1_2"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/><a NAME="E11:3_1_1_2"/>
<font color="Maroon">x</font>,<font color="Maroon">a</font> <a href="pencil_1.html#R1">are_collinear</a> 
 <b>by </b><i>, , , <a class="ref" href="pencil_1.html#D3">PENCIL_1:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">l</font> being   <a href="pencil_1.html#NM1">Block</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E13:3_1_1_2"/><i><font color="Green">E55</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">a</font></span><a href="tarski.html#K2">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">l</font>
 <b>by </b><i>, <a class="ref" href="pencil_1.html#D1">PENCIL_1:def 1</a></i>;<br/><a NAME="E14:3_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">l</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">l</font> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T38">ZFMISC_1:38</a></i>;<br/><b>hence </b><a NAME="E15:3_1_1_2"/>
 ex <font color="Olive">l</font> being  <a href="pencil_1.html#NM1">Block</a> of <font color="Maroon">S</font> st <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">l</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:3"/>
<font color="Maroon">S</font> is <a href="pencil_1.html#V8">without_isolated_points</a>
 <b>by </b><i><a class="ref" href="pencil_1.html#D9">PENCIL_1:def 9</a></i>;<br/>


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